11 Search Results for "Chiesa, Alessandro"


Document
On Parallel Repetition of PCPs

Authors: Alessandro Chiesa, Ziyi Guan, and Burcu Yıldız

Published in: LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)


Abstract
Parallel repetition refers to a set of valuable techniques used to reduce soundness error of probabilistic proofs while saving on certain efficiency measures. Parallel repetition has been studied for interactive proofs (IPs) and multi-prover interactive proofs (MIPs). In this paper we initiate the study of parallel repetition for probabilistically checkable proofs (PCPs). We show that, perhaps surprisingly, parallel repetition of a PCP can increase soundness error, in fact bringing the soundness error to one as the number of repetitions tends to infinity. This "failure" of parallel repetition is common: we find that it occurs for a wide class of natural PCPs for NP-complete languages. We explain this unexpected phenomenon by providing a characterization result: the parallel repetition of a PCP brings the soundness error to zero if and only if a certain "MIP projection" of the PCP has soundness error strictly less than one. We show that our characterization is tight via a suitable example. Moreover, for those cases where parallel repetition of a PCP does bring the soundness error to zero, the aforementioned connection to MIPs offers preliminary results on the rate of decay of the soundness error. Finally, we propose a simple variant of parallel repetition, called consistent parallel repetition (CPR), which has the same randomness complexity and query complexity as the plain variant of parallel repetition. We show that CPR brings the soundness error to zero for every PCP (with non-trivial soundness error). In fact, we show that CPR decreases the soundness error at an exponential rate in the repetition parameter.

Cite as

Alessandro Chiesa, Ziyi Guan, and Burcu Yıldız. On Parallel Repetition of PCPs. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


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@InProceedings{chiesa_et_al:LIPIcs.ITCS.2024.34,
  author =	{Chiesa, Alessandro and Guan, Ziyi and Y{\i}ld{\i}z, Burcu},
  title =	{{On Parallel Repetition of PCPs}},
  booktitle =	{15th Innovations in Theoretical Computer Science Conference (ITCS 2024)},
  pages =	{34:1--34:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-309-6},
  ISSN =	{1868-8969},
  year =	{2024},
  volume =	{287},
  editor =	{Guruswami, Venkatesan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.34},
  URN =		{urn:nbn:de:0030-drops-195629},
  doi =		{10.4230/LIPIcs.ITCS.2024.34},
  annote =	{Keywords: probabilistically checkable proofs, parallel repetition}
}
Document
Hardness of Approximation for Stochastic Problems via Interactive Oracle Proofs

Authors: Gal Arnon, Alessandro Chiesa, and Eylon Yogev

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)


Abstract
Hardness of approximation aims to establish lower bounds on the approximability of optimization problems in NP and beyond. We continue the study of hardness of approximation for problems beyond NP, specifically for stochastic constraint satisfaction problems (SCSPs). An SCSP with 𝗄 alternations is a list of constraints over variables grouped into 2𝗄 blocks, where each constraint has constant arity. An assignment to the SCSP is defined by two players who alternate in setting values to a designated block of variables, with one player choosing their assignments uniformly at random and the other player trying to maximize the number of satisfied constraints. In this paper, we establish hardness of approximation for SCSPs based on interactive proofs. For 𝗄 ≤ O(log n), we prove that it is AM[𝗄]-hard to approximate, to within a constant, the value of SCSPs with 𝗄 alternations and constant arity. Before, this was known only for 𝗄 = O(1). Furthermore, we introduce a natural class of 𝗄-round interactive proofs, denoted IR[𝗄] (for interactive reducibility), and show that several protocols (e.g., the sumcheck protocol) are in IR[𝗄]. Using this notion, we extend our inapproximability to all values of 𝗄: we show that for every 𝗄, approximating an SCSP instance with O(𝗄) alternations and constant arity is IR[𝗄]-hard. While hardness of approximation for CSPs is achieved by constructing suitable PCPs, our results for SCSPs are achieved by constructing suitable IOPs (interactive oracle proofs). We show that every language in AM[𝗄 ≤ O(log n)] or in IR[𝗄] has an O(𝗄)-round IOP whose verifier has constant query complexity (regardless of the number of rounds 𝗄). In particular, we derive a "sumcheck protocol" whose verifier reads O(1) bits from the entire interaction transcript.

Cite as

Gal Arnon, Alessandro Chiesa, and Eylon Yogev. Hardness of Approximation for Stochastic Problems via Interactive Oracle Proofs. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 24:1-24:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{arnon_et_al:LIPIcs.CCC.2022.24,
  author =	{Arnon, Gal and Chiesa, Alessandro and Yogev, Eylon},
  title =	{{Hardness of Approximation for Stochastic Problems via Interactive Oracle Proofs}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{24:1--24:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.24},
  URN =		{urn:nbn:de:0030-drops-165867},
  doi =		{10.4230/LIPIcs.CCC.2022.24},
  annote =	{Keywords: hardness of approximation, interactive oracle proofs, stochastic satisfaction problems}
}
Document
Interactive Oracle Proofs of Proximity to Algebraic Geometry Codes

Authors: Sarah Bordage, Mathieu Lhotel, Jade Nardi, and Hugues Randriam

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)


Abstract
In this work, we initiate the study of proximity testing to Algebraic Geometry (AG) codes. An AG code C = C(𝒳, 𝒫, D) over an algebraic curve 𝒳 is a vector space associated to evaluations on 𝒫 ⊆ 𝒳 of functions in the Riemann-Roch space L_𝒳(D). The problem of testing proximity to an error-correcting code C consists in distinguishing between the case where an input word, given as an oracle, belongs to C and the one where it is far from every codeword of C. AG codes are good candidates to construct probabilistic proof systems, but there exists no efficient proximity tests for them. We aim to fill this gap. We construct an Interactive Oracle Proof of Proximity (IOPP) for some families of AG codes by generalizing an IOPP for Reed-Solomon codes, known as the FRI protocol [Eli Ben-Sasson et al., 2018]. We identify suitable requirements for designing efficient IOPP systems for AG codes. Our approach relies on a neat decomposition of the Riemann-Roch space of any invariant divisor under a group action on a curve into several explicit Riemann-Roch spaces on the quotient curve. We provide sufficient conditions on an AG code C that allow to reduce a proximity testing problem for C to a membership problem for a significantly smaller code C'. As concrete instantiations, we study AG codes on Kummer curves and curves in the Hermitian tower. The latter can be defined over polylogarithmic-size alphabet. We specialize the generic AG-IOPP construction to reach linear prover running time and logarithmic verification on Kummer curves, and quasilinear prover time with polylogarithmic verification on the Hermitian tower.

Cite as

Sarah Bordage, Mathieu Lhotel, Jade Nardi, and Hugues Randriam. Interactive Oracle Proofs of Proximity to Algebraic Geometry Codes. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 30:1-30:45, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


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@InProceedings{bordage_et_al:LIPIcs.CCC.2022.30,
  author =	{Bordage, Sarah and Lhotel, Mathieu and Nardi, Jade and Randriam, Hugues},
  title =	{{Interactive Oracle Proofs of Proximity to Algebraic Geometry Codes}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{30:1--30:45},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.30},
  URN =		{urn:nbn:de:0030-drops-165923},
  doi =		{10.4230/LIPIcs.CCC.2022.30},
  annote =	{Keywords: Algebraic geometry codes, Interactive oracle proofs of proximity, Proximity testing}
}
Document
Interactive Proofs for Verifying Machine Learning

Authors: Shafi Goldwasser, Guy N. Rothblum, Jonathan Shafer, and Amir Yehudayoff

Published in: LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)


Abstract
We consider the following question: using a source of labeled data and interaction with an untrusted prover, what is the complexity of verifying that a given hypothesis is "approximately correct"? We study interactive proof systems for PAC verification, where a verifier that interacts with a prover is required to accept good hypotheses, and reject bad hypotheses. Both the verifier and the prover are efficient and have access to labeled data samples from an unknown distribution. We are interested in cases where the verifier can use significantly less data than is required for (agnostic) PAC learning, or use a substantially cheaper data source (e.g., using only random samples for verification, even though learning requires membership queries). We believe that today, when data and data-driven algorithms are quickly gaining prominence, the question of verifying purported outcomes of data analyses is very well-motivated. We show three main results. First, we prove that for a specific hypothesis class, verification is significantly cheaper than learning in terms of sample complexity, even if the verifier engages with the prover only in a single-round (NP-like) protocol. Moreover, for this class we prove that single-round verification is also significantly cheaper than testing closeness to the class. Second, for the broad class of Fourier-sparse boolean functions, we show a multi-round (IP-like) verification protocol, where the prover uses membership queries, and the verifier is able to assess the result while only using random samples. Third, we show that verification is not always more efficient. Namely, we show a class of functions where verification requires as many samples as learning does, up to a logarithmic factor.

Cite as

Shafi Goldwasser, Guy N. Rothblum, Jonathan Shafer, and Amir Yehudayoff. Interactive Proofs for Verifying Machine Learning. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 41:1-41:19, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


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@InProceedings{goldwasser_et_al:LIPIcs.ITCS.2021.41,
  author =	{Goldwasser, Shafi and Rothblum, Guy N. and Shafer, Jonathan and Yehudayoff, Amir},
  title =	{{Interactive Proofs for Verifying Machine Learning}},
  booktitle =	{12th Innovations in Theoretical Computer Science Conference (ITCS 2021)},
  pages =	{41:1--41:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-177-1},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{185},
  editor =	{Lee, James R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.41},
  URN =		{urn:nbn:de:0030-drops-135806},
  doi =		{10.4230/LIPIcs.ITCS.2021.41},
  annote =	{Keywords: PAC learning, Fourier analysis of boolean functions, Complexity gaps, Complexity lower bounds, Goldreich-Levin algorithm, Kushilevitz-Mansour algorithm, Distribution testing}
}
Document
On Local Testability in the Non-Signaling Setting

Authors: Alessandro Chiesa, Peter Manohar, and Igor Shinkar

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)


Abstract
Non-signaling strategies are a generalization of quantum strategies that have been studied in physics for decades, and have recently found applications in theoretical computer science. These applications motivate the study of local-to-global phenomena for non-signaling functions. We prove that low-degree testing in the non-signaling setting is possible, assuming that the locality of the non-signaling function exceeds a threshold. We additionally show that if the locality is below the threshold then the test fails spectacularly, in that there exists a non-signaling function which passes the test with probability 1 and yet is maximally far from being low-degree. Along the way, we present general results about the local testability of linear codes in the non-signaling setting. These include formulating natural definitions that capture the condition that a non-signaling function "belongs" to a given code, and characterizing the sets of local constraints that imply membership in the code. We prove these results by formulating a logical inference system for linear constraints on non-signaling functions that is complete and sound.

Cite as

Alessandro Chiesa, Peter Manohar, and Igor Shinkar. On Local Testability in the Non-Signaling Setting. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 26:1-26:37, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{chiesa_et_al:LIPIcs.ITCS.2020.26,
  author =	{Chiesa, Alessandro and Manohar, Peter and Shinkar, Igor},
  title =	{{On Local Testability in the Non-Signaling Setting}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{26:1--26:37},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.26},
  URN =		{urn:nbn:de:0030-drops-117112},
  doi =		{10.4230/LIPIcs.ITCS.2020.26},
  annote =	{Keywords: non-signaling strategies, locally testable codes, low-degree testing, Fourier analysis}
}
Document
On the Impossibility of Probabilistic Proofs in Relativized Worlds

Authors: Alessandro Chiesa and Siqi Liu

Published in: LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)


Abstract
We initiate the systematic study of probabilistic proofs in relativized worlds, where the goal is to understand, for a given oracle, the possibility of "non-trivial" proof systems for deterministic or nondeterministic computations that make queries to the oracle. This question is intimately related to a recent line of work that seeks to improve the efficiency of probabilistic proofs for computations that use functionalities such as cryptographic hash functions and digital signatures, by instantiating them via constructions that are "friendly" to known constructions of probabilistic proofs. Informally, negative results about probabilistic proofs in relativized worlds provide evidence that this line of work is inherent and, conversely, positive results provide a way to bypass it. We prove several impossibility results for probabilistic proofs relative to natural oracles. Our results provide strong evidence that tailoring certain natural functionalities to known probabilistic proofs is inherent.

Cite as

Alessandro Chiesa and Siqi Liu. On the Impossibility of Probabilistic Proofs in Relativized Worlds. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 57:1-57:30, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)


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@InProceedings{chiesa_et_al:LIPIcs.ITCS.2020.57,
  author =	{Chiesa, Alessandro and Liu, Siqi},
  title =	{{On the Impossibility of Probabilistic Proofs in Relativized Worlds}},
  booktitle =	{11th Innovations in Theoretical Computer Science Conference (ITCS 2020)},
  pages =	{57:1--57:30},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-134-4},
  ISSN =	{1868-8969},
  year =	{2020},
  volume =	{151},
  editor =	{Vidick, Thomas},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.57},
  URN =		{urn:nbn:de:0030-drops-117420},
  doi =		{10.4230/LIPIcs.ITCS.2020.57},
  annote =	{Keywords: probabilistically checkable proofs, relativization}
}
Document
Probabilistic Checking Against Non-Signaling Strategies from Linearity Testing

Authors: Alessandro Chiesa, Peter Manohar, and Igor Shinkar

Published in: LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)


Abstract
Non-signaling strategies are a generalization of quantum strategies that have been studied in physics over the past three decades. Recently, they have found applications in theoretical computer science, including to proving inapproximability results for linear programming and to constructing protocols for delegating computation. A central tool for these applications is probabilistically checkable proofs (PCPs) that are sound against non-signaling strategies. In this paper we prove that the exponential-length constant-query PCP construction due to Arora et al. (JACM 1998) is sound against non-signaling strategies. Our result offers a new length-vs-query tradeoff when compared to the non-signaling PCP of Kalai, Raz, and Rothblum (STOC 2013 and 2014) and, moreover, may serve as an intermediate step to a proof of a non-signaling analogue of the PCP Theorem.

Cite as

Alessandro Chiesa, Peter Manohar, and Igor Shinkar. Probabilistic Checking Against Non-Signaling Strategies from Linearity Testing. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 25:1-25:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)


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@InProceedings{chiesa_et_al:LIPIcs.ITCS.2019.25,
  author =	{Chiesa, Alessandro and Manohar, Peter and Shinkar, Igor},
  title =	{{Probabilistic Checking Against Non-Signaling Strategies from Linearity Testing}},
  booktitle =	{10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
  pages =	{25:1--25:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{124},
  editor =	{Blum, Avrim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.25},
  URN =		{urn:nbn:de:0030-drops-101188},
  doi =		{10.4230/LIPIcs.ITCS.2019.25},
  annote =	{Keywords: probabilistically checkable proofs, linearity testing, non-signaling strategies}
}
Document
Testing Linearity against Non-Signaling Strategies

Authors: Alessandro Chiesa, Peter Manohar, and Igor Shinkar

Published in: LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018)


Abstract
Non-signaling strategies are collections of distributions with certain non-local correlations. They have been studied in Physics as a strict generalization of quantum strategies to understand the power and limitations of Nature's apparent non-locality. Recently, they have received attention in Theoretical Computer Science due to connections to Complexity and Cryptography. We initiate the study of Property Testing against non-signaling strategies, focusing first on the classical problem of linearity testing (Blum, Luby, and Rubinfeld; JCSS 1993). We prove that any non-signaling strategy that passes the linearity test with high probability must be close to a quasi-distribution over linear functions. Quasi-distributions generalize the notion of probability distributions over global objects (such as functions) by allowing negative probabilities, while at the same time requiring that "local views" follow standard distributions (with non-negative probabilities). Quasi-distributions arise naturally in the study of Quantum Mechanics as a tool to describe various non-local phenomena. Our analysis of the linearity test relies on Fourier analytic techniques applied to quasi-distributions. Along the way, we also establish general equivalences between non-signaling strategies and quasi-distributions, which we believe will provide a useful perspective on the study of Property Testing against non-signaling strategies beyond linearity testing.

Cite as

Alessandro Chiesa, Peter Manohar, and Igor Shinkar. Testing Linearity against Non-Signaling Strategies. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 17:1-17:37, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


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@InProceedings{chiesa_et_al:LIPIcs.CCC.2018.17,
  author =	{Chiesa, Alessandro and Manohar, Peter and Shinkar, Igor},
  title =	{{Testing Linearity against Non-Signaling Strategies}},
  booktitle =	{33rd Computational Complexity Conference (CCC 2018)},
  pages =	{17:1--17:37},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-069-9},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{102},
  editor =	{Servedio, Rocco A.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.17},
  URN =		{urn:nbn:de:0030-drops-88731},
  doi =		{10.4230/LIPIcs.CCC.2018.17},
  annote =	{Keywords: property testing, linearity testing, non-signaling strategies, quasi-distributions}
}
Document
Proofs of Proximity for Distribution Testing

Authors: Alessandro Chiesa and Tom Gur

Published in: LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)


Abstract
Distribution testing is an area of property testing that studies algorithms that receive few samples from a probability distribution D and decide whether D has a certain property or is far (in total variation distance) from all distributions with that property. Most natural properties of distributions, however, require a large number of samples to test, which motivates the question of whether there are natural settings wherein fewer samples suffice. We initiate a study of proofs of proximity for properties of distributions. In their basic form, these proof systems consist of a tester that not only has sample access to a distribution but also explicit access to a proof string that depends on the distribution. We refer to these as NP distribution testers, or MA distribution testers if the tester is a probabilistic algorithm. We also study the more general notion of IP distribution testers, in which the tester interacts with an all-powerful untrusted prover. We investigate the power and limitations of proofs of proximity for distributions and chart a landscape that, surprisingly, is significantly different from that of proofs of proximity for functions. Our main results include showing that MA distribution testers can be quadratically stronger than standard distribution testers, but no stronger than that; in contrast, IP distribution testers can be exponentially stronger than standard distribution testers, but when restricted to public coins they can be at best quadratically stronger.

Cite as

Alessandro Chiesa and Tom Gur. Proofs of Proximity for Distribution Testing. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 53:1-53:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)


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@InProceedings{chiesa_et_al:LIPIcs.ITCS.2018.53,
  author =	{Chiesa, Alessandro and Gur, Tom},
  title =	{{Proofs of Proximity for Distribution Testing}},
  booktitle =	{9th Innovations in Theoretical Computer Science Conference (ITCS 2018)},
  pages =	{53:1--53:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-060-6},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{94},
  editor =	{Karlin, Anna R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.53},
  URN =		{urn:nbn:de:0030-drops-83114},
  doi =		{10.4230/LIPIcs.ITCS.2018.53},
  annote =	{Keywords: distribution testing, proofs of proximity, property testing}
}
Document
On Axis-Parallel Tests for Tensor Product Codes

Authors: Alessandro Chiesa, Peter Manohar, and Igor Shinkar

Published in: LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)


Abstract
Many low-degree tests examine the input function via its restrictions to random hyperplanes of a certain dimension. Examples include the line-vs-line (Arora, Sudan 2003), plane-vs-plane (Raz, Safra 1997), and cube-vs-cube (Bhangale, Dinur, Livni 2017) tests. In this paper we study tests that only consider restrictions along axis-parallel hyperplanes, which have been studied by Polishchuk and Spielman (1994) and Ben-Sasson and Sudan (2006). While such tests are necessarily "weaker", they work for a more general class of codes, namely tensor product codes. Moreover, axis-parallel tests play a key role in constructing LTCs with inverse polylogarithmic rate and short PCPs (Polishchuk, Spielman 1994; Ben-Sasson, Sudan 2008; Meir 2010). We present two results on axis-parallel tests. (1) Bivariate low-degree testing with low-agreement. We prove an analogue of the Bivariate Low-Degree Testing Theorem of Polishchuk and Spielman in the low-agreement regime, albeit with much larger field size. Namely, for the 2-wise tensor product of the Reed-Solomon code, we prove that for sufficiently large fields, the 2-query variant of the axis-parallel line test (row-vs-column test) works for arbitrarily small agreement. Prior analyses of axis-parallel tests assumed high agreement, and no results for such tests in the low-agreement regime were known. Our proof technique deviates significantly from that of Polishchuk and Spielman, which relies on algebraic methods such as Bezout's Theorem, and instead leverages a fundamental result in extremal graph theory by Kovari, Sos, and Turan. To our knowledge, this is the first time this result is used in the context of low-degree testing. (2) Improved robustness for tensor product codes. Robustness is a strengthening of local testability that underlies many applications. We prove that the axis-parallel hyperplane test for the m-wise tensor product of a linear code with block length n and distance d is Omega(d^m/n^m)-robust. This improves on a theorem of Viderman (2012) by a factor of 1/poly(m). While the improvement is not large, we believe that our proof is a notable simplification compared to prior work.

Cite as

Alessandro Chiesa, Peter Manohar, and Igor Shinkar. On Axis-Parallel Tests for Tensor Product Codes. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 39:1-39:22, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


Copy BibTex To Clipboard

@InProceedings{chiesa_et_al:LIPIcs.APPROX-RANDOM.2017.39,
  author =	{Chiesa, Alessandro and Manohar, Peter and Shinkar, Igor},
  title =	{{On Axis-Parallel Tests for Tensor Product Codes}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)},
  pages =	{39:1--39:22},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-044-6},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{81},
  editor =	{Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.39},
  URN =		{urn:nbn:de:0030-drops-75882},
  doi =		{10.4230/LIPIcs.APPROX-RANDOM.2017.39},
  annote =	{Keywords: tensor product codes, locally testable codes, low-degree testing, extremal graph theory}
}
Document
Interactive Oracle Proofs with Constant Rate and Query Complexity

Authors: Eli Ben-Sasson, Alessandro Chiesa, Ariel Gabizon, Michael Riabzev, and Nicholas Spooner

Published in: LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)


Abstract
We study interactive oracle proofs (IOPs) [BCS16,RRR16], which combine aspects of probabilistically checkable proofs (PCPs) and interactive proofs (IPs). We present IOP constructions and techniques that enable us to obtain tradeoffs in proof length versus query complexity that are not known to be achievable via PCPs or IPs alone. Our main results are: 1. Circuit satisfiability has 3-round IOPs with linear proof length (counted in bits) and constant query complexity. 2. Reed-Solomon codes have 2-round IOPs of proximity with linear proof length and constant query complexity. 3. Tensor product codes have 1-round IOPs of proximity with sublinear proof length and constant query complexity. For all the above, known PCP constructions give quasilinear proof length and constant query complexity [BS08,Din07]. Also, for circuit satisfiability, [BKKMS13] obtain PCPs with linear proof length but sublinear (and super-constant) query complexity. As in [BKKMS13], we rely on algebraic-geometry codes to obtain our first result; but, unlike that work, our use of such codes is much "lighter" because we do not rely on any automorphisms of the code. We obtain our results by proving and combining "IOP-analogues" of tools underlying numerous IPs and PCPs: * Interactive proof composition. Proof composition [AS98] is used to reduce the query complexity of PCP verifiers, at the cost of increasing proof length by an additive factor that is exponential in the verifier's randomness complexity. We prove a composition theorem for IOPs where this additive factor is linear. * Sublinear sumcheck. The sumcheck protocol [LFKN92] is an IP that enables the verifier to check the sum of values of a low-degree multi-variate polynomial on an exponentially-large hypercube, but the verifier's running time depends linearly on the bound on individual degrees. We prove a sumcheck protocol for IOPs where this dependence is sublinear (e.g., polylogarithmic). Our work demonstrates that even constant-round IOPs are more efficient than known PCPs and IPs.

Cite as

Eli Ben-Sasson, Alessandro Chiesa, Ariel Gabizon, Michael Riabzev, and Nicholas Spooner. Interactive Oracle Proofs with Constant Rate and Query Complexity. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 40:1-40:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)


Copy BibTex To Clipboard

@InProceedings{bensasson_et_al:LIPIcs.ICALP.2017.40,
  author =	{Ben-Sasson, Eli and Chiesa, Alessandro and Gabizon, Ariel and Riabzev, Michael and Spooner, Nicholas},
  title =	{{Interactive Oracle Proofs with Constant Rate and Query Complexity}},
  booktitle =	{44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)},
  pages =	{40:1--40:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-041-5},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{80},
  editor =	{Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.40},
  URN =		{urn:nbn:de:0030-drops-74713},
  doi =		{10.4230/LIPIcs.ICALP.2017.40},
  annote =	{Keywords: probabilistically checkable proofs, interactive proofs, proof composition, sumcheck}
}
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